Applying DDM on a Relational Matching task

# Applying DDM on a Relational Matching task
### Adriana Felisa Chávez De la Peña
### COGS 237; Spring 2022
### May 27, 2022

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# Week 8 Presentation

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# First, a recap...

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## Recap: The task

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## Recap: The data

- 82 participants

- 27 trials per participant

- 12 Relational trials
  - 15 Control trials
  
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- 6 textures `$\times$` 6 shapes.

- Time constraint on response (2800ms or 3500ms)

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## Recap: Drift Diffusion Model

Sequential-sampling model for "rapid" binary decision making.

![](data:image/png;base64,#./Pictures/DDM_Illustration3.png)

(Johnson et al., 2017)
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`$$Y_{i} = (\mbox{Choice}_{i},\mbox{RT}_{i})$$`

`$$Y_{i} \sim \mbox{Wiener}(\alpha,\beta,\tau,\delta)$$`

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# Hierarchical DDM

(Let me take a step back this time...)

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## Vandekerckhove, Tuerlinckx & Lee, 2011

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## Vandekerckhove, Tuerlinckx & Lee, 2011

`$$y_{pij} \sim Wiener(\alpha_{p},\beta,\tau_{pij},\delta_{pij})$$`

`$$\beta = 0.5$$`

`$$\tau_{pij} \sim Normal(\theta_p,X_p^2)$$`

`$$\delta_{pij} \sim Normal(\nu_{pi}, \eta_{p}^2)$$`

`$$\alpha_p \sim Normal(\mu_\alpha,\sigma_\alpha^2)$$`

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`$$\theta_p \sim Normal(\mu_\theta,\sigma_\theta^2)$$`

`$$\nu_p \sim Normal(\mu_{\nu_i},\sigma_{\nu_i}^2)$$`

`$$X_p \sim Normal(\mu_x,\sigma_x^2)$$`
`$$\eta_p \sim Normal (\mu_\eta,\sigma_\eta^2)$$`

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## Vandekerckhove, Tuerlinckx & Lee, 2011

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## Vandekerckhove, Tuerlinckx & Lee, 2011

```r
model {

# Defining population distributions.
 # Alpha = Boundary distance
 mu.alpha ~ dunif(0.02,0.30)
 sigma.alpha ~ dunif(0.0001,0.15)
 prec.alpha <- pow(sigma.alpha,-2)
 # Theta = Mean ND-RT
 mu.theta ~ dunif(0.02,0.70)
 sigma.theta ~ dunif(0.0001,0.15)
 prec.theta <- pow(sigma.theta,-2)
 # Ji = Variance on ND-RT
 mu.chi ~ dunif(0.0001,0.15)
 sigma.chi ~ dunif(0.0001,0.10)
 prec.chi <- pow(sigma.chi,-2)
 # Eta = Variance on drift rate
 mu.eta ~ dunif(0.001,0.45)
 sigma.eta ~ dunif(0.0001,0.15)
 prec.eta <- pow(sigma.eta,-2)
```
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.panel[.panel-name[Condition-level]

```r
# Condition-level distribution

for(i in 1:nc){
 # Nu = Mean drift rate
 mu.nu[i] ~ dunif(-.5,.6)
 sigma.nu[i] ~ dunif(0,0.6)
 prec.nu[i] <- pow(sigma.nu[i],-2)
 }
```
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.panel[.panel-name[Participant-level]

```r
# Participant-level distributions

for(p in 1:np){
 # Alpha = Boundary distance
 alpha[p] ~ dnorm(mu.alpha,prec.alpha)I(0.01,0.25)
 # Starting point = alpha*beta
 zeta_init[p] <- alpha[p]*beta
 # Theta = Mean ND-RT
 theta[p] ~ dnorm(mu.theta,prec.theta)I(0.01,0.80)
 # Ji = Variance on ND-RT
 chi[p] ~ dnorm(mu.chi,prec.chi)I(0.00001,0.49)
 prec.tau[p] <- pow(chi[p],-2)
 
 # Eta = Variance on drift rate
 eta[p] ~ dnorm(mu.eta,prec.eta)I(0.00001,0.49)
 
 # For each participant, there's a condition-level distribution
 for(i in 1:nc){
 # Nu = Mean drift rate
 nu[p,i] ~ dnorm(mu.nu[i],prec.nu[i])I(-.7,.7)
 }}
```
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.panel[.panel-name[Likelihood]

```r
  for(j in 1:N){
      # Trial-by-trial variability in ND-RT.
      tau[j] ~ dnorm(theta[pnum[j]],prec.tau[pnum[j]])I(0,)
    
      # Each observation comes from a Wiener distribution with parameters...
      t[j] ~ dwiener.eta(alpha[pnum[j]],       # alpha = boundary distance    
                         tau[j],               # tau = ND-RT
                         zeta_init[pnum[j]],   # alpha*beta = Starting point
                         nu[pnum[j],cond[j]],  # Mean drift rate
                         eta[pnum[j]])         # Variance drift rate
  }}
```

The code is available <a href="https://supp.apa.org/psycarticles/supplemental/a0021765/met_vandekerckhove_0110_supp2.pdf" target="_blank">here</a>.

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## Week 5 proposal

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## Quick revisit

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## Our version of the model.

- The standard deviation of the drift rate can be fixed, (Ratcliff & Rouder, 1998;  `$s=0.1$`)

- Let's assume a single (but still, **unknown**) value for `$\eta$`

`$$\delta_{pij} \sim Normal(\nu_{pi}, \eta)$$`
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- We substitute the Normal distributions on `$\tau$` and `$\alpha$` for a Uniform distribution.

`$$\tau_{pij} \sim Uniform(\tau_p^{low},\tau_p^{high})$$`

`$$\alpha_{pij} \sim Uniform(\alpha_p^{low},\alpha_p^{high})$$`

- All participants, tasks and trials are influenced by a single `$\beta$`, which is not fixed, but rather:

`$$\beta \sim Uniform(0.4,0.7)$$`

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## Our model

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## Posterior distributions

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# Thank you!